Nice applications of Cartan-Hadamards theorem?

Question

I was wondering if anyone knew of some nice/suprising results that made use of Cartan-Hadamards theorem (or its generalizations).

With kind regards

Answer 1

There is a version of Cartan-Hadamard theorem involving somewhat singular metric spaces. The most remarkable application I know appears in a series of papers by Burago, Ferliger and Kononenko, mostly written here:

Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. (2) 147 (1998), no. 3, 695–708.

From the Math review of this paper:

This is a remarkable paper—it solves a long-standing and celebrated open problem in the theory of billiard dynamical systems and mechanics. The authors prove that in a gas of N hard balls in the open space the number of possible collisions is uniformly bounded (until now, the problem had been only solved for N=3). The authors give an explicit upper bound for the number of collisions between N hard balls of arbitrary masses. They also solve a more general billiard problem: for multidimensional semidispersing billiards (i.e. with walls concave inward) the number of collisions near any "nondegenerate'' corner point is uniformly bounded. A simple new criterion of nondegeneracy of a corner point is found. The authors give an elementary and very elegant solution of the above problems. In addition, they generalized the result (and the proof) to billiards on Riemannian manifolds with bounded sectional curvature, where the particle moves along geodesics between elastic collisions with walls. This involves the theory of Aleksandrov spaces.

See also:

D. Burago, S. Ferleger, A. Kononenko, A geometric approach to semi-dispersing billiards. Hard ball systems and the Lorentz gas, 9–27, Encyclopaedia Math. Sci., 101, Math. Phys., II, Springer, Berlin, 2000.

for a somewhat informal discussion of the results and methods.